Structure of Banach algebras over locally compact groups

局部紧群上的 Banach 代数的结构

• 批准号: RGPIN-2014-05354
• 负责人: Samei, Ebrahim
• 金额: $ 0.8万
• 依托单位: University of Saskatchewan
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611586
• 关键词: Structure; Banach; algebras; over; locally

My main field of research is harmonic analysis with an emphasis on non-commutative groups such as amenable groups and Lie groups. Harmonic analysis is a branch of mathematics concerned with the representation of functions, signals, or waves using Fourier series and Fourier transforms. It has become a vast subject with applications in areas such as signal processing and quantum mechanics over the past two centuries. The research projects that I pursue primarily relate to more modern branches of harmonic analysis and is about the understanding of the structure of various Banach algebras related to locally compact groups, including group algebras and Fourier algebras. My objective is to study in a deeper level the structure of these algebras and their relation to the representations of the groups. One aspect of the structure theory of Banach algebras over locally compact groups I am interested in deals with cohomological properties of these algebras. The seminal paper in this respect is certainly B. E. Johnson’s memoir in 1972 in which he introduces and explores the concept of amenability for a Banach algebra. Z-J Ruan’s fundamental work in 1995 of introducing the concept of operator amenability, through the theory of operator spaces, and applying it in non-commutative harmonic analysis has also had a significant influence on this area. An important open problem in this area is the characterization of the weak amenability of the Fourier algebra on non-compact connected groups (the compact and some non-compact cases have already been established). One research project I plan to pursue is to investigate a more general property, namely the failure of spectral synthesis of the antidiagonal for non-abelian, non-compact connected groups. It is expected that this will imply the failure of the weak amenability and provide more such examples. I will also look at the “local behavior” of their Hochschild cohomology. The overall goal is to get more information about the structure of the algebras or the groups using these results. Another research project of mine deals with "non-communicative" analogue of Beurling algebras. These algebras have played important roles in different areas of harmonic analysis. They are L^1-algebras associated to locally compact groups when we put extra "weights" on the groups. With my co-author H. H. Lee, we developed the corresponding "dual theory" and created the Beurling-Fourier algebras (shortly BF algebras). In several co-authored research projects, I have shown that in some aspect, they behave similar to the classical Beurling algebras whereas in others, they behave very differently. However, this theory is very recent and subsequently there is still much research to be completed concerning them. From one direction, more detailed study of BF algebras on compact groups must be conducted. We also need to investigate BF algebras on non-compact, non-abelian groups e.g. Heisenberg groups and the ax+b-group. The research I am planning to pursue using my NSERC Discovery grant is all a continuation of my work in the past 10 years. I am happy to report that the results obtained so far have been substantial and have received positive reviews from other experts in the fields. Canada is growing as one of the research-intensive countries in the world, and to make sure we continue as such, we need to not only maintain our research programs but also grow and expand them. One necessary means, among many others, is to have faculties with strong research initiatives and programs in all areas of science and technologies including mathematics. The research projects I have outlined above will be part of many streams of research being conducted across the country. This will have a great and positive impact in our future.

我的主要研究领域是谐波分析，重点是非交换群，如可调群和李群。谐波分析是用傅立叶级数和傅立叶变换表示函数、信号或波的数学分支。在过去的两个世纪里，它已经成为一个广泛的学科，在信号处理和量子力学等领域得到了应用。我所从事的研究项目主要涉及调和分析的现代分支，是关于理解与局部紧群相关的各种巴拿赫代数的结构，包括群代数和傅立叶代数。我的目标是在更深层次上研究这些代数的结构以及它们与群表示的关系。